A great variety of electronic sensors, ranging from sensors used in astronomical studies to sensors used for monitoring physiological parameters of the human body, provide electrical signals representing desired measurements. Beyond the realm of astronomy and medicine, electronic sensors now pervade in all aspects of our lives and throughout our environment.
In data collection, an electronic converter, such as an analog to digital converter (“ADC”) generally receives an electrical signal from one or more sensors and produces digital data at a regular rate (discrete data points in time) representing each converted measurement value. Such streams of digital data can be displayed on a graph as data points plotted in time illustrating how data changes in the “time domain”. In a time domain representation, the horizontal or independent axis represents time.
Where signals or aspects of signals, such as physiological signals, change at almost periodic rates, such as the time interval between successive heart beats as measured using an electrocardiogram, it can also be of interest to transform such non-uniformly spaced in time data to the “frequency domain”. In a frequency domain representation, the horizontal or independent axis represents frequency, while the vertical axis is a complex number representing the amplitude and phase of the corresponding frequency. Frequency domain graphs are commonly presented in a form known as a “power spectrum”, a graph showing the squared amplitude of each estimated frequency over a range of frequencies and ignoring the phase. The mathematics used to convert or transform data acquired at uniformly spaced intervals in time from the time domain to the frequency domain is well understood. The method of choice for such conversion or transformation from time domain to frequency domain is generally the discrete Fourier transform (“DFT”) most often accomplished by a relatively fast mathematical algorithm known as the Fast Fourier Transform (“FFT”).
The DFT and FFT routines can provide a means of transformation for optimal circumstances of data collection (e.g. data recorded accurately at uniform intervals of time and with no missing data points). However, DFT and FFT routines perform poorly where points are missing in time or where data sampling does not occur with uniform spacing. In many fields of data collection, data points can be occasionally missing or can occur at non-uniform periods in time. For example a human heart beat generally has a non-uniform rhythm reflecting both a range of normal human activities as well as, in some cases, heart disorders. Also, in astronomical studies, it is common place for data to be missing due to signals falling below detection levels or signals lost in noise.
One existing technique interpolates or “guesses” what missing data points should have been, based on earlier and later acquired points. The “interpolated” data points are “forced” into the missing or uniformly spaced data point positions to form a corrected data set now suitable for transformation by FFT. Because interpolation techniques “manufacture” missing data points or apply mathematical processes to move data points to uniform time spacings in order to apply an FFT or DFT routine, they can unintentionally create a data set that does not produce an accurate result in the frequency domain.
There are methods that can directly perform a transformation to a power spectrum without first performing an interpolation. The “Lomb-Scargle Transform” (“LST”), for example, was developed to analyze astronomical non-uniformly spaced data in time and missing data points. The resulting LST power spectrum is also known as a Lomb-Scargle periodogram. Unfortunately, the LST requires a large number of mathematical operations and therefore fast processing speeds. There has been some success in producing streamlined methods for the LST (analogous to how the FFT streamlines DFT calculations), however the streamlined LST techniques are still computationally intensive. Also, the LST is an “off-line” computational technique. In an off-line computational method, a large amount of computation must be repeated for new data points and new frequency transformations that overlap in time with previous data points and transformations. Off-line transformations are generally less suitable for “real-time” signal processing. For example, many types of medical equipment, such as ECG monitors used for heart rate monitoring, need to produce continual results as new data is being received. Existing LST frequency transformation techniques estimate only power in the frequency domain, and do not generate phase information.
What is needed is a method that requires far less computational resources than the existing LST related methods, that can generate phase information in the frequency domain, and that is suitable for real-time signal processing.